0$ (see also Remark 3.2 at the end of
section 3).
To state Theorem \ref{thm1}, let us first recall the well-known fact that
(\ref{H1}) implies the existence of a
sequence $\mu_i(h)$ of eigenvalues of the linear problem
$$
-\Delta u=\mu h(x)u, \quad u\in {\cal D}^{1,2}(\mathbb{R}^N)$$
such that $0 < \mu_1(h) < \mu_2(h)\leqslant \mu_3(h)\leqslant \ldots \to
\infty$. This is due to the fact that
the map
$u\mapsto \int hu^2$ is compact in ${\cal D}^{1,2}(\mathbb{R}^N)$.
\begin{theorem} \label{thm1}
Consider problem {\rm (\ref{Plambda})}, suppose that
$a(x)$ changes sign and that {\rm (\ref{H1})-(\ref{G4})} hold. If
$\mu_k(h) \Lambda$ (see
\cite{CT}), it is natural to ask whether other
(possibly sign-changing) solutions exist for any $\lambda>0$. Theorem \ref{thm1}
answers this question in the affirmative
whenever $\lambda$ is not one of the eigenvalues $\mu_j(h)$. It should also
be noted that, in contrast with
the assumptions made in \cite{CT} (see also the references therein), we
assume that the function $a(x)$ satisfies
condition (\ref{A2}) instead of having a ``thick" zero set (see Lemma 1.2 in
\cite{CT}) and, for Theorem \ref{thm1}, we do not
assume that $g(s)$ behaves like a superlinear power at zero. Condition
(\ref{A2}) was introduced in \cite{BCN} and later used
in \cite{DR,RTT}.
Typically, Theorem \ref{thm1} applies to nonlinearities such as, for example,
$g(s)=|s|^{p-2}s+\theta(s)|s|^{q-2}s$ where
$2 0,\quad \forall s\in \mathbb{R},\; s\neq 0, \label{G3'}\\
\lim_{s\to 0}\frac{sg(s)}{|s|^q}=\ell_0\in (0,\infty)\quad
\mbox{ for some } q>0 \label{G4'}
\end{gather}
and prove the following.
\begin{theorem} \label{thm2}
Consider problem {\rm (\ref{Plambda})}, suppose that
$a(x)$ changes sign, $g$ is odd and that {\rm (\ref{A1})-(\ref{G2})} and
{\rm (\ref{H1'})-(\ref{G4'})} hold. Then, for any
$\lambda \in \mathbb{R}$, problem {\rm (\ref{Plambda})} has
infinitely many solutions.
\end{theorem}
We observe that similar conclusions for {\em bounded} domains $\Omega$ were
obtained in \cite{AT1} for odd $g$ and in \cite{B,T1} for
perturbations of an odd function $g$, under a
different set of assumptions (again, the ``thick zero set" assumption on
$a(x)$ was used by those authors). Concerning
(\ref{H1'}), we mention that the assumption on the positivity can be replaced
by the weaker assumption that $h>0$ a.e. over
the bounded open set $\{ x: a(x)>0\}$ (see Lemma \ref{epsilon-estimate} in
section 2).
The proofs of Theorems \ref{thm1} and \ref{thm2} are given in sections
3 and 4, respectively. In order to prove our results, we have to
face the lack of compactness due to the unboundedness of the domain and the
fact that $a(x)$ changes sign. We overcome
this by constructing an appropriate sequence of solutions of the equation in
(\ref{Plambda}), lying in $H_0^1(\Omega_n)$, where
$\Omega_n=B_{R_n}(0)\subset
\mathbb{R}^N$ with $R_n\to \infty$; the estimates on the Morse indices of these
solutions \cite{DR} insure the boundedness
of the sequence and allows us to take its limit in the space ${\cal
D}^{1,2}(\mathbb{R}^N)$ (weak limit, in case of Theorem \ref{thm1};
strong limit, in case of Theorem 2). Roughly speaking, our argument relies
on the fact that the Morse index
estimates provide Palais-Smale sequences for
(\ref{Plambda}) with the additional property that the sequence is bounded in
$L^{\infty}(\mathbb{R}^N)$; together with a
version of Brezis-Lieb lemma proved in section 2, this yields compactness
for the problem. Regarding the
multiplicity result, it will follow from the observation that, under the
assumptions of Theorem 2, these Palais-Smale
sequences can be constructed at arbitrarily large levels of energy.
We mention that, although our results in section 2 suggest that perhaps one
could work directly in a convenient Banach
subspace of the Hilbert space ${\cal D}^{1,2}(\mathbb{R}^N)$, we prefer to use the
approximated sequence of Hilbert spaces
$H_0^1(\Omega_n)$. This is mainly because, in the latter case, Morse index
estimates in Hilbert spaces and their
connections with blow-up techniques (see \cite{DR}) can be directly applied
to our problem without the need of
additional theoretical developments.
\section{Preliminary results}
We recall that we denote by ${\cal D}^{1,2}(\mathbb{R}^N)$ the closure of ${\cal
D}(\mathbb{R}^N)$ with respect
to the norm
$\|u\|=(\int_{\mathbb{R}^N}|\nabla u|^2)^{1/2}$. The notation $\|u\|_r$
($1\leqslant r\leqslant \infty$) stands for the norm in
$L^r$ spaces and the Sobolev continuous immersion of ${\cal D}^{1,2}(\mathbb{R}^N)$
into $L^{2^{\star}}(\mathbb{R}^N)$ will be
repeatedly used (see \cite{W}).
As mentioned in the Introduction, we prove Theorems 1 and 2 through an
approximation argument in bounded open balls of
$\mathbb{R}^N$. Under assumption (\ref{H1}), we denote by $(\mu_i(h))_{i\in \mathbb{N}}$ and
$(\mu_i^R(h))_{i\in \mathbb{N}}$ (for each $R>0$) the
sequence of eigenvalues of the problems
\begin{gather*}
-\Delta u =\mu h(x)u, \quad u\in {\cal D}^{1,2}(\mathbb{R}^N) \quad \mbox{and }\\
-\Delta u =\mu^R h(x)u, \quad u\in H_0^1(B_R(0)).
\end{gather*}
\begin{lemma}\label{eigenvalues}
Given $k\in \mathbb{N}$ and $\varepsilon >0$ there
exists $R_0>0$ such that
$$
|\mu_k(h)-\mu_k^R(h)| < \varepsilon,\quad \forall R\geqslant R_0.$$
\end{lemma}
\paragraph{Proof.} We first recall that the theory of compact symmetric operators on
Hilbert spaces implies the following
variational characterization of $\mu_k(h)$:
$$
\mu_k(h)= \min_{{\rm dim}X=k}\max_{u\in X,u\neq 0}
\frac{\int_{\mathbb{R}^N}|\nabla u|^2}{\int_{\mathbb{R}^N}hu^2}
= \max_{u\in X_k,u\neq 0} \frac{\int_{\mathbb{R}^N}|\nabla
u|^2}{\int_{\mathbb{R}^N}hu^2}\,,
$$
where we denote by $X_k$ the eigenspace associated with the first $k$
eigenvalues and $X$ runs through the
$k$-dimensional subspaces of ${\cal D}^{1,2}(\mathbb{R}^N)$. A similar formula
holds for $\mu_k^R(h)$ where, now, $X$ is a
subspace of
$H_0^1(B_R(0))$. Since $H_0^1(B_R(0))\subset {\cal D}^{1,2}(\mathbb{R}^N)$, this
shows in particular that $\mu_k(h) \leqslant
\mu_k^R(h)$.
On the other hand, let $X_k$ be spanned by $\{ \varphi_1,\ldots,\varphi_k\}$
and denote by $\Psi_R$ a smooth function
$\Psi_R\in {\cal D}(\mathbb{R}^N)$ such that $0\leqslant \Psi_R \leqslant 1$,
$\Psi_R=1$ in $B_R(0)$, $\Psi_R=0$ in
$\mathbb{R}^N\setminus B_{2R}(0)$ and $\|\nabla \Psi_R\|_{\infty}\leqslant CR^{-1}$
for every $R>0$. By unique continuation,
the space spanned by $\{ \Psi_R\varphi_1,\ldots,\Psi_R\varphi_k\}$ has
dimension $k$. Therefore, in view of
the above variational characterizations, the lemma will be proved if we
show that, for every large $R$,
\begin{equation}\label{eigenvalues-1}
\frac{\int|\nabla( u\Psi_R)|^2}{\int h(u\Psi_R)^2} \leqslant \varepsilon +
\frac{\int|\nabla u|^2}{\int h u^2}, \quad \forall u \in X_k.
\end{equation}
Except otherwise indicated, all integrals are taken over the whole space
$\mathbb{R}^N$. On the other hand,
(\ref{eigenvalues-1}) will follow once we show that
\begin{equation}\label{eigenvalues-2}
\frac{\int|\nabla( u\Psi_R)|^2}{\int h(u\Psi_R)^2} -
\frac{\int|\nabla u|^2}{\int h u^2}\to 0 \quad \mbox{ as } \quad R\to
\infty,
\end{equation}
uniformly for $u\in X_k$, $\int|\nabla u|^2 =1$. In order to prove
(\ref{eigenvalues-2}), let $R_n$ be any sequence such that $R_n \to \infty$,
denote $\Psi_n=\Psi_{R_n}$ and let $u_n\in
X_k$ be any sequence such that $\int|\nabla u_n|^2 =1$. Since $X_k$ is
finite dimensional,
$$
\varepsilon_n:=\int_{\mathbb{R}^N\setminus B_{R_n}(0)} |u_n|^{2^{\star}}\to 0.
$$
Similarly,
$$
\int |\nabla u_n|^2 \Psi_n^2 \to 1 \quad \mbox{ and } \quad \liminf_{n\to
\infty} (\int hu_n^2\; \int h u_n^2\Psi_n^2)>0.
$$
Recalling that $\int|\nabla u_n|^2 =1$, it remains to prove that
\begin{equation}\label{eigenvalues-3}
\int|\nabla( u_n\Psi_n)|^2\; \int h u_n^2 - \int h u_n^2\Psi_n^2 \to 0.
\end{equation}
Observing that, by H\"older inequality,
$$
\int hu_n^2(1-\Psi_n^2)\leqslant \int_{\mathbb{R}^N\setminus B_{R_n}(0)}hu_n^2
\leqslant \|h\|_{N/2} \;\varepsilon_n^{2/2^{\star}}\to 0,
$$
the expression in (\ref{eigenvalues-3}) can be written as
$$
(\int|\nabla( u_n\Psi_n)|^2-1)\; \int h u_n^2 +\mbox{o}(1),$$
where $\mbox{o}(1)\to 0$ as $n\to \infty$. Finally, we observe that
\begin{align*}
|\int |\nabla (u_n\Psi_n)|^2-1|\leqslant & \mbox{o}(1) + \int u_n^2 |\nabla
\Psi_n|^2 + 2 \int |u_n|\,|\Psi_n|\, |\nabla
u_n|\,|\nabla \Psi_n|\\
\leqslant & \mbox{o}(1) + CR_n^{-2} \int_{\mathbb{R}^N\setminus
B_{R_n}(0)}u_n^2\\
&+ 2
(R_n^{-2} \int_{\mathbb{R}^N\setminus B_{R_n}(0)}u_n^2)^{1/2}\; (\int|\nabla
u_n|^2)^{1/2}\\
\leqslant & \mbox{o}(1) + C' (\varepsilon_n^{2/2^{\star}} +
\varepsilon_n^{1/2^{\star}})\to 0
\end{align*}
and the lemma follows.\hfill$\Box$\smallskip
For any $R>0$ and $k\in \mathbb{N}$ we denote by $X_{k,R}$ the closure in
$H_0^1(B_R(0))$ of the eigenspaces associated with
the eigenvalues $\mu_i^R(h)$ for $i\geqslant k+1$.
\begin{lemma}\label{epsilon-estimate}
Assume {\rm (\ref{H1'})} and let $p\in [1,2^{\star})$. Given
$\varepsilon >0$ and
$k_1\in \mathbb{N}$ there exist
$R_0 >0$ and $k\in \mathbb{N}$, $k\geqslant k_1$,
such that
$$
\int_{B_1(0)}|u|^{p} \leqslant \varepsilon \Big( \int_{B_R(0)} |\nabla u|^2
\Big)^{p/2} \quad
\forall R\geqslant R_0 \, \forall u \in X_{k,R}.$$
\end{lemma}
\paragraph{Proof.}
Assuming the contrary, there exist sequences $k_n\to \infty$, $R_n
\to \infty$, $u_n\in X_{k_n,R_n}$ such that
$\int_{B_{R_n}(0)} |\nabla u_n|^2 =1$ and $\int_{B_1(0)}|u_n|^{p} \geqslant
\varepsilon$. Up to a subsequence,
we may assume that $(u_n)$ converges weakly in ${\cal D}^{1,2}(\mathbb{R}^N)$ to
some function $u$ and that $u_n \to u$
strongly in
$L^{p}(B_1(0))$. In particular, $\int_{B_1(0)}|u|^{p}\geqslant \varepsilon$.
Since $u_n\in X_{k_n,R_n}$, it
follows from the definition of $\mu_{k_n}^{R_n}(h)$ that
\begin{equation}\label{definition-of-eigenvalues}
1=\int_{B_{R_n}(0)} |\nabla u_n|^2\geqslant \mu_{k_n}^{R_n}(h)
\int_{B_{R_n}(0)} hu_n^2.
\end{equation}
On the other hand, as observed at the beginning of the proof of Lemma
\ref{eigenvalues}, for every $n$ we have that
\begin{equation}\label{monotonicity-of-eigenvalues}
\mu_{k_n}^{R_n}(h)\geqslant \mu_{k_n}(h)\to \infty.
\end{equation}
Combining this with (\ref{definition-of-eigenvalues}) yields that
$$
\int_{B_1(0)} hu^2 = \lim_{n\to \infty} \int_{B_1(0)} hu_n^2\leqslant
\lim_{n\to \infty} \int_{B_{R_n}(0)} hu_n^2=0.
$$
Since $h>0$ in $B_1(0)$, this implies $u=0$ in $B_1(0)$, contradicting the
fact that
$\int_{B_1(0)}|u|^{p}\geqslant \varepsilon$.\hfill$\Box$\smallskip
We end this section with a version of the well-known Brezis-Lieb lemma which
is suitable for our purposes. We first
recall the following.
\begin{lemma}[Brezis-Lieb lemma]
Let $H: \mathbb{R}\to\mathbb{R}$ be continuous, $H\geqslant 0$ and satisfy
\begin{equation}\label{a-kind-of-convexity}
\forall \varepsilon \; \exists C_{\varepsilon}:\; |H(s+t)-H(s)|\leqslant
\varepsilon|H(s)|+C_{\varepsilon}|H(t)|,\quad \forall s,t\in \mathbb{R}.
\end{equation}
For a given sequence $(w_n)$ of measurable functions in $\mathbb{R}^N$, suppose
$w_n \to w$ a.e., $\sup_n\int H(w_n) 0$ for all $s\neq 0$ and
suppose that for some $0

0 \quad \mbox{ and } \quad
\lim_{|s|\to \infty}\frac{H(s)}{|s|^p} =\ell_{\infty} >0.
$$
Then $H$ satisfies condition {\rm (\ref{a-kind-of-convexity})}.
\end{proposition}
\paragraph{Proof.}
Step 1. Given $\varepsilon >0$, fix $0 < \varepsilon_0 < R_0$ and
$C_2/C_1<1+\varepsilon$, $C'_2/C'_1<1+\varepsilon$,
in such a way that
\begin{gather*}
C_1|s|^q\leqslant H(s) \leqslant C_2|s|^q, \quad \forall |s| \leqslant
2\varepsilon_0,
C'_1|s|^p\leqslant H(s) \leqslant C'_2|s|^p, \quad \forall |s| \geqslant
R_0.
\end{gather*}
Then condition (\ref{a-kind-of-convexity}) is trivially satisfied in case
$|t|\leqslant
\varepsilon_0$ and $|s|\leqslant \varepsilon_0$; and also in case
$|t|\geqslant R_0$, $|s|\geqslant R_0$ and
$|t+s|\geqslant R_0$.
\noindent Step 2. Take $\lambda>0$ such that
\begin{equation}\label{tricky-estimate-1}
H(t)\geqslant \lambda, \quad \forall t: \; \varepsilon_0 \leqslant
|t|\leqslant R_0.
\end{equation}
Next, choose $\delta \in ]0,\varepsilon_0[$ in such a way that
\begin{gather}\label{tricky-estimate-2}
|H(s+t)-H(s)| \leqslant \varepsilon\lambda, \quad \forall |s|\leqslant
R_0,\; \forall |t|\leqslant \delta, \\
\label{tricky-estimate-3}
|H(s+t)-H(s)| \leqslant \varepsilon H(s), \quad \forall |s|\geqslant
R_0,\; \forall |t|\leqslant \delta.
\end{gather}
\noindent Step 3. We prove condition (\ref{a-kind-of-convexity}) in the case
$|t|\leqslant \delta$. As mentioned in step 1 above, we may
assume that $|s|\geqslant \varepsilon_0$. Thus, if $|s|\geqslant
R_0$ the conclusion follows from
(\ref{tricky-estimate-3}) while if $|s|\leqslant R_0$ the conclusion follows
from
(\ref{tricky-estimate-1}) and (\ref{tricky-estimate-2}).
\noindent Step 4. It remains to check the case where $|t|\geqslant \delta$. First, we
observe that there exists $C_{\delta}>0$ and $C_3>0$ such that
\begin{equation}\label{tricky-estimate-4}
C_{\delta}|t|^p \leqslant H(t), \; \forall |t|\geqslant \delta \mbox{ and }
\;
|H(t)|=H(t)\leqslant C_3(1+|t|^p), \; \forall t\in \mathbb{R}.
\end{equation}
Suppose then that $|t|\geqslant \delta$. In case $|s|\leqslant R_0+|t|$ we
deduce from (\ref{tricky-estimate-4}) that
$$
|H(s+t)-H(s)| \leqslant H(s+t)+H(s) \leqslant C_4(1+|t|^p) \leqslant
C_{\varepsilon}H(t),$$
for some large constant $C_{\varepsilon} >0$. Finally, in case $|s|\geqslant
R_0 +|t|$ we have that $|s|\geqslant R_0$,
$|s+t|\geqslant R_0$ and we can argue as in step 1, thanks to the first
inequality in (\ref{tricky-estimate-4}).\hfill$\Box$
\section{Proof of Theorem \ref{thm1}}
Throughout this section we assume that the conditions in Theorem \ref{thm1} are
satisfied. As a consequence of Lemma
\ref{eigenvalues}, it follows from the assumptions of Theorem 1 that we can
fix $R$ so large that the constant $\lambda$
appearing in (\ref{Plambda}) satisfies, for every large $R>0$,
\begin{equation}\label{nonresonance}
\mu_k^R(h) < \lambda < \mu_{k+1}^R(h).
\end{equation}
Take any sequence $R_n\to \infty$ and let $\Omega_n=B_{R_n}(0)$. We denote
by $I$ the functional
\begin{equation}\label{energy-functional}
I(u)=\frac{1}{2}\int(|\nabla u|^2-\lambda h(x)u^2) - \int a(x)G(u).
\end{equation}
Under assumptions (\ref{H1}), (\ref{A1}), (\ref{G1}), (\ref{G2}) and for each $n\in \mathbb{N}$, $I$ is a
$C^2$ functional over $H_0^1(\Omega_n)$ and
its critical points in $H_0^1(\Omega_n)$ correspond to solutions of
(\ref{Plambda}) lying in $H_0^1(\Omega_n)$. For any
such critical point $u$, we denote by $m(u)$ the Morse index of $u$ with
respect to $I$, that is, the supremum of the
dimensions of the linear subspaces of $H_0^1(\Omega_n)$ on which the
quadratic form $D^2I(u)$ is negative definite.
\begin{proposition}\label{bounded-domain}
Under the assumptions of Theorem \ref{thm1}, for
every $n$ the equation in {\rm (\ref{Plambda})} has
a nonzero solution $u_n\in H_0^1(\Omega_n)$ and the following holds:\\[1mm]
{\rm (i)} $(u_n)$ is bounded in $L^{\infty}(\Omega_n)$.\\[1mm]
{\rm (ii)} Either $m(u_n)\leqslant k-1$ for every $n$ or else $\limsup_{n\to
\infty}I(u_n) >0$.
\end{proposition}
\paragraph{Proof.} Step 1. Since the proof is based on \cite{DR,RTT}, we shall be sketchy. At
first we observe that the existence of
nonzero solutions $(u_n)$ follows straightforwardly from the main theorem in
\cite{RTT}, which was proved by means of a
truncation argument and the use of a critical point theorem in
\cite{LL,LW}. We point out that, since $a(x)$ changes
sign, the truncation argument is needed in order to insure the Palais-Smale
condition for the functional $I$ over
$H_0^1(\Omega_n)$ as well as to obtain the required geometric condition on
$I$. At this point, assumptions (\ref{A1}),
(\ref{G3}) and (\ref{G4}) are not used; regarding the unique continuation property
mentioned in Lemma 2 of
\cite{RTT}, we also observe that the equation
$-\Delta u =\mu h(x) u$ can be written as $Ku=u/\mu$ where
$Ku=(-\Delta)^{-1}(h(x)u)$ in $H_0^1(\Omega_n)$, so that the
proof of the quoted lemma remains unchanged.
\noindent Step 2. By construction, the sequence of the Morse indices of these solutions is
bounded ($m(u_n)\leqslant k$ for every $n$,
see \cite{DR,RTT}). Also, our regularity assumptions imply that $u_n\in
C(\overline{\Omega_n})\cap C^2(\Omega_n)$.
Assume by contradiction that
$$
M_n:=\|u_n\|_{\infty}=\max_{\Omega_n}u_n=u_n(x_n)\to +\infty$$
for some $x_n \in \Omega_n$ (the case where
$\|u_n\|_{\infty}=\max_{\Omega_n}(-u_n)$ is similar). Since $\Delta u_n(x_n)
\leqslant 0$, the equation in (\ref{Plambda}) shows that
$$
a^{-}(x_n)g(M_n) \leqslant CM_n + a^+(x_n)g(M_n),$$
where we denoted $a^+:=\max\{a,0\}$ and $a^-:=\max\{-a,0\}$. From assumption
(\ref{A1}) we see that $(x_n)$ is bounded. Thus,
up to a subsequence, we can assume that
$x_n\to x_0 \in \mathbb{R}^N$ and $a(x_0)\geqslant 0$.
At this point, the blow-up argument in
section 3 of \cite{RTT} can be applied, leading
to a contradiction. Indeed, since $m(u_n)\leqslant k$ and $\|u_n\|_{\infty}
\to \infty$, it is shown in
\cite{RTT} that the sequence $v_n(x)=u_n(\lambda_n x+x_n)/M_n$, with
$\lambda_n=M_n^{(2-p)/2}$ or
$\lambda_n=M_n^{(2-p)/3}$ depending on whether $a(x_0)>0$ or $a(x_0)=0$,
respectively, converges uniformly in compact
sets to 0. Since, by definition, $v_n(0)=1$, this is impossible and
therefore part $(i)$ in
Proposition \ref{bounded-domain} is proved.
\noindent Step 3. Finally, as explained in Proposition 2 of \cite{DR}, each
solution $u_n$ can be chosen in such a way that either
$m(u_n)\leqslant k-1$ or else, for some small $r_n>0$,
\begin{equation}\label{energy-level-estimate}
I(u_n)\geqslant \inf\{ I(u): \; u\in X_{k,n},\;\|u\|=r_n\},
\end{equation}
where we denote by $X_{k,n}$ the closure of the eigenspaces associated with
the eigenvalues $\mu_i^{R_n}(h)$ for
$i\geqslant k+1$. Actually, by the construction in \cite{DR},
(\ref{energy-level-estimate}) holds for a modified
functional
$$
\widetilde{I}(u)=\frac{1}{2}\int(|\nabla u|^2-\lambda h(x)u^2) - \int
a^+(x)G(u)+ \int a^-(x)\widetilde{G}(u),$$
where $\widetilde{G}$ is a truncation of the function $G$ which still
satisfies (\ref{G3}). Thus, (\ref{energy-level-estimate})
should be written as
\begin{equation}\label{true-energy-level-estimate}
I(u_n)\geqslant \inf\{ \widetilde{I}(u): \; u\in X_{k,n},\;\|u\|=r_n\}.
\end{equation}
So, in order to prove $(ii)$ in Proposition \ref{bounded-domain} it is
enough to show that the
right hand side of
(\ref{true-energy-level-estimate}) can be bounded below by some positive
constant which does not depend on $n$. Now,
to show this, let $u$ be any function in $X_{k,n}$. From
(\ref{nonresonance}) we see that, for some constant $\eta >0$
independent of $n$,
\begin{align*}
\widetilde{I}(u) & \geqslant \eta \|u\|^2 - \int a^+G(u)+ \int
a^-(x)\widetilde{G}(u) \\
& \geqslant \eta \|u\|^2 - \int a^+G(u)
\end{align*}
where we have used (\ref{G3}).
To estimate the above integral term, we use the fact that (\ref{G1})
and (\ref{G2}) imply that $|G(s)|\leqslant \varepsilon
s^2 +C_{\varepsilon} |s|^{p}$ for any $\varepsilon >0$. As a consequence,
and since $a^+$ has compact
support (cf. (\ref{A1})), we obtain
$$
\int a^+u^2 \leqslant C(\int_{\{a>0\}} |u|^{2^{\star}})^{2/2^{\star}}
\leqslant
C(\int_{\Omega_n} |u|^{2^{\star}})^{2/2^{\star}} \leqslant
C'\|u\|^2,$$
and a similar estimate for $\int a^{+}|u|^p$. Therefore,
\begin{equation}\label{the-energy-is-bounded-below}
\widetilde{I}(u)
\geqslant \|u\|^2 \;(\eta - \varepsilon C'-C'_{\varepsilon}\|u\|^{p-2}),
\end{equation}
where $C'$ and $C'_{\varepsilon}$ are independent of $n$. By choosing
$\varepsilon$ small we see that we can select
$r_n$ independently of $n$. This proves the claim and completes the proof
of the proposition.\hfill$\Box$\smallskip
Now we can complete the proof of Theorem \ref{thm1}.
\paragraph{Proof of Theorem \ref{thm1} completed.}
Let $(u_n)$ be given by
Proposition \ref{bounded-domain}. If we multiply the
equation in (\ref{Plambda}) by $u_n$ and integrate over $\Omega_n$ we obtain
\begin{equation}\label{multiply-and-integrate}
\int |\nabla u_n|^2+ \int a^-g(u_n)u_n = \lambda \int h u_n^2 + \int a^+
g(u_n)u_n.
\end{equation}
Except othervise indicated, all integrals are taken over $\mathbb{R}^N$, by
extending the solutions as zero outside $\Omega_n$.
Since $(\|u_n\|_{\infty})$ is bounded, it follows from
(\ref{multiply-and-integrate}) and (\ref{A1}) that
\begin{equation}\label{consequence-of-multiply-and-integrate}
\int |\nabla u_n|^2+ \int g(u_n)u_n \leqslant \lambda \int h u_n^2 + C.
\end{equation}
We claim that $(\|u_n\|)$ is bounded. Indeed, suppose $t_n:=\|u_n\|\to
\infty$ and denote $v_n:=u_n/t_n$. Up to a
subsequence, $v_n\to v$ weakly in ${\cal D}^{1,2}(\mathbb{R}^N)$ and a.\ e. Fix any
function $\varphi \in {\cal
D}(\mathbb{R}^N)$. If we multiply the
equation in (\ref{Plambda}) by $\frac{au_n\varphi}{t_n^2}$ and integrate we
see that
\begin{equation}\label{multiply-and-integrate-by-au}
\int a^2\; \frac{g(u_n)u_n}{t_n^2}\; \varphi \leqslant C',
\end{equation}
for some positive constant $C'$ depending on $\varphi$. Since $g$ is
superlinear at infinity (cf. (\ref{G2})), we deduce
from (\ref{multiply-and-integrate-by-au}) that $a^2|v|\varphi=0$. Since
$\varphi$ is arbitrary and since $a$ vanishes
on a set of measure zero, we obtain that $v=0$. Going back to
(\ref{consequence-of-multiply-and-integrate}) and using
the fact that the map $u\mapsto \int hu^2$ is compact in ${\cal
D}^{1,2}(\mathbb{R}^N)$, we conclude that $\|v_n\|^2\to 0$ as
$n\to
\infty$, which is a contradiction. This proves that $(\|u_n\|)$ is bounded.
Up to a subsequence, let $u$ be a weak limit of $(u_n)$ in ${\cal
D}^{1,2}(\mathbb{R}^N)$ such that $u_n \to u$
a.\ e. Using test functions in (\ref{Plambda}) we see that $u$ is a solution
of (\ref{Plambda}). It
remains to show that $u \neq 0$.
Suppose by contradiction that $(u_n)$ converges weakly to 0 in ${\cal
D}^{1,2}(\mathbb{R}^N)$. Since $a^+$ has compact support,
it follows from (\ref{multiply-and-integrate}) and $(\ref{G1})-(\ref{G2})$ that
$$
\int |\nabla u_n|^2+ \int a^-g(u_n)u_n \to 0
$$
and subsequently, using (\ref{A1}), that
\begin{equation}\label{strongly-to-zero}
\int |\nabla u_n|^2\to 0 \quad \mbox{ and } \quad \int |a|\;g(u_n)u_n \to
0.
\end{equation}
Using $(\ref{G3})-(\ref{G4})$, this implies that
\begin{equation}\label{energy-to-zero}
I(u_n)=\frac{1}{2}(\int |\nabla u_n|^2 -\lambda \int hu_n^2) - \int aG(u_n)
\to 0
\end{equation}
as $n\to \infty$. So, Proposition \ref{bounded-domain} $(ii)$ implies that
$m(u_n)\leqslant k-1$ for
every $n$.
On the other hand, we have seen in the proof of Lemma \ref{eigenvalues} (see
(\ref{eigenvalues-1})) that, since $\lambda
>\mu_k(h)$, there exist a $k$-dimensional space $X\subset H_0^1(B_R(0))$
(for
some large $R>0$) and a constant $\eta >0$
such that
\begin{equation}\label{large-Morse-index-1}
I''(0)(v,v) = \int |\nabla v|^2-\lambda \int h v^2 \leqslant -2\eta \int
|\nabla v|^2,\quad \forall v\in X.
\end{equation}
Observe also that, by elliptic regularity, $X\subset L^{\infty}(B_R(0))$.
Therefore, since $u_n\to 0$ a.e.\ and
$(\|u_n\|_{\infty})$ is bounded, Lebesgue's dominated convergence theorem
implies that
\begin{equation}\label{large-Morse-index-2}
\int g'(u_n)v^2 \to 0, \quad \mbox{ uniformly in } v\in X: \; \int |\nabla
v|^2=1.
\end{equation}
Combining (\ref{large-Morse-index-1}) and (\ref{large-Morse-index-2}) we
conclude that, for large $n$,
$$
I''(u_n)(v,v) = \int |\nabla v|^2-\lambda \int h v^2 -\int a g'(u_n)v^2
\leqslant -\eta \int |\nabla v|^2,\quad \forall
v\in X.$$
By definition, this says that $m(u_n)\geqslant k$ for large $n$, which
contradicts the fact that $m(u_n)\leqslant k-1$
and concludes the proof of Theorem \ref{thm1}.\hfill$\Box$
\begin{remark} \label{rmk3.2} \rm
Under the assumptions of Theorem \ref{thm1}, suppose $a(x)$ is
bounded. In this case, it is sufficient to
assume that the inequality in (\ref{G4}) holds for all $|s|\leqslant \delta$,
for some $\delta >0$. Indeed, if $a(x)$ is
bounded the conclusion in (\ref{energy-to-zero}) is a consequence of the
fact that $\int |\nabla u_n|^2\to 0$, $\int
g(u_n)u_n\to 0$ (as follows from (\ref{strongly-to-zero})) and
\begin{eqnarray*}
\int |G(u_n)|&=&\int_{\{|u_n|\leqslant \delta\}}
G(u_n)+\int_{\{|u_n|\geqslant \delta\}} G(u_n)\nonumber\\
&\leqslant& C\int g(u_n)u_n + C_{\delta}\int |u_n|^{2^{\star}}\to 0.
\end{eqnarray*}
\end{remark}
\begin{remark} \label{rmk3.3}\rm
In the same manner, one can also treat the case where
the equation in (\ref{Plambda})
contains an extra term of the form $-b(x)|u|^{r-2}u$ with $b\geqslant 0$,
$b$ bounded and $r>2$ small with respect to $p$.
\end{remark}
More generally, following \cite{DR}, the conclusion of Theorem \ref{thm1} holds for
an equation of the form
$$
-\Delta u = \lambda h(x) u + a(x) g(u)-b(x)f(u),$$
where $\lambda$, $h$, $a$, $g$ are as in Theorem \ref{thm1} and $b\geqslant 0$, $b\in
C^1\cap L^{\infty}(\mathbb{R}^N)$, $f$ satisfies
assumptions similar to (\ref{G1}), (\ref{G2}), (\ref{G4}) and, moreover,
$$
|f'(s)| \leqslant C|s|^{r-2}\quad \mbox{and} \quad f(s)s-r\int_0^sf
\leqslant Cs^2, \quad \forall |s|\geqslant 1,$$
where $C>0$ and $20$ we consider the modified problem
\begin{equation}
-\Delta u = \lambda h(x) u + a^+(x)g(u) - a^-(x)g_j(u), \quad u\in
H_0^1(B_R(0)),\label{PjR}
\end{equation}
where $a^{\pm}:=\max\{\pm a,0\}$. The corresponding energy functional is
{\em even} and is given by
$$
I^j_R(u)=\frac{1}{2}
\int_{B_R(0)} ( |\nabla u|^2 - \lambda h(x)u^2 )
- \int_{B_R(0)} a^+(x) G(u) + \int_{B_R(0)} a^-(x) G_j(u)$$
for $u \in H^1_0({B_R(0)})$. For any $\ell\in \mathbb{N}$ and $R>0$, we have the
orthogonal sum
$$
H^1_0({B_R(0)})= V_{\ell,R}\oplus X_{\ell,R},$$
where $V_{\ell,R}$ stands for the $\ell$-dimensional eigenspace associated
with the first $\ell$ eigenvalues
$\mu_i^R(h)$,
$i=1,\ldots,\ell$. Finally, for any critical point $u$ of $I^j_R$ we denote
by $m^j_R(u)$ its Morse index.
Recall that $I^j_R$ satisfies the Palais-Smale condition over
$H^1_0({B_R(0)})$ if every sequence $(u_n)\subset
H^1_0({B_R(0)})$ such that $(I^j_R(u_n))$ is bounded and
$\|\nabla I^j_R(u_n)\| \to 0$ has a convergent subsequence.
The next lemma collects some facts that were proved in \cite[Prop.3]{RTT}.
\begin{lemma}\label{results-from-DR}
Assume {\rm (\ref{H1}), (\ref{A2})}, and {\rm (\ref{G2})}. Then
\begin{enumerate}
\item[(a)] For any $j\in \mathbb{N}$ and $R>0$, the functional $I^j_R$ satisfies
the Palais-Smale condition over
$H^1_0({B_R(0)})$.
\item[(b)] For any $j,\ell \in \mathbb{N}$ and $R>0$,
$I^j_R(u) \to -\infty$ as $\|u\|\to \infty$, $u\in V_{\ell,R}$.
\item[(c)] For any $\ell \in \mathbb{N}$ and $R>0$ there exist $j_0\in \mathbb{N}$ and
$c>0$ such that $\|u\|_{\infty}\leqslant c$ for
every $j\geqslant j_0$ and every critical point $u$ of $I^j_R$ such that
$m^j_R(u)\leqslant \ell$.
\end{enumerate}
\end{lemma}
Observe that (c) is an a-priori estimate in $L^{\infty}(B_R(0))$ for the
solutions of $(P)^{j}_{R}$ having bounded Morse
index; for each fixed $R$, the estimate depends on $R$ but {\em not} on
$j$.
Next, for every $j,\ell\in \mathbb{N}$ and $R>0$ we let
\begin{equation}\label{definition-of-b^k}
b^j_{\ell,R}=\inf \{ I^j_R(u): \; u\in X_{\ell-1,R},\; \|u\|=r_\ell\},
\end{equation}
where $r_\ell$ is a large constant to be chosen later.
\begin{lemma}\label{estimate-from-below}
Assume {\rm (\ref{H1'}), (\ref{A1}), (\ref{G1})--(\ref{G3})}, and let
$d\in \mathbb{R}$. Then there
exist $\ell\in \mathbb{N}$ and $R_0 >0$ such that
$$
b^j_{\ell,R} \geqslant d,\quad \forall j\in \mathbb{N}\;\; \forall R\geqslant
R_0.$$
\end{lemma}
\paragraph{Proof.}
The proof is similar to the one in (\ref{true-energy-level-estimate})
except that we now use Lemma \ref{epsilon-estimate}.
Without loss of generality (cf. (\ref{A1})) we assume that
$\{ x: a(x)>0\} \subset B_1(0)$. At first
we recall from (\ref{monotonicity-of-eigenvalues}) that we can fix
$\ell_1\in \mathbb{N}$ such that
$\mu_{\ell}^R(h) > \lambda$ for every $R>0$ and every $\ell\geqslant
\ell_1$, so that, for some $\eta >0$,
$$
\int_{B_R(0)} |\nabla u|^2 - \lambda\int_{B_R(0)} hu^2 \geqslant \eta
\|u\|^2, \quad \forall \ell\geqslant \ell_1 \;
\forall R>0\; \forall u\in X_{\ell-1,R}.$$
Since, by (\ref{G2}), $\int a^{+}G(u) \leqslant c\left(
\int_{B_1(0)}|u|^p+1\right)$,
we deduce that, for some positive constants $c_1$, $c_2$,
\begin{equation}\label{the-energy-is-uniformly-bounded-below}
I^j_R(u) \geqslant c_1(c_2\|u\|^2 - \int_{B_1(0)}|u|^{p}-1),
\end{equation}
for all $ j\in \mathbb{N}$, $R>0$, $\ell\geqslant \ell_1$ and $u\in X_{\ell-1,R}$.
Fix $\epsilon >0$ so small that
\begin{equation}\label{r-is-large-1}
c_2-\varepsilon r^{p-2}=\frac{c_2}{2}
\end{equation}
where $r$ is given by
\begin{equation}\label{r-is-large-2}
c_1c_2\frac{r^2}{2} -c_1\geqslant d.
\end{equation}
For this value of $\varepsilon$, let $\ell\geqslant \ell_1$ and $R_0$ be
given by Lemma \ref{epsilon-estimate}. Thanks
to that lemma, we can rewrite (\ref{the-energy-is-uniformly-bounded-below})
as
\begin{equation}\label{the-energy-is-indeed-uniformly-bounded-below}
I^j_R(u) \geqslant c_1 \|u\|^2(c_2 - \varepsilon \|u\|^{p-2})-c_1,\quad
\forall j\in \mathbb{N}\; \forall R\geqslant R_0\; \forall u\in X_{\ell-1,R}.
\end{equation}
From
(\ref{r-is-large-1})--(\ref{the-energy-is-indeed-uniformly-bounded-below})
and the definition in
(\ref{definition-of-b^k}) with $r_{\ell}=r$ we conclude that
$$
b^j_{\ell,R} \geqslant c_1 r^2 (c_2-\varepsilon
r^{p-2})-c_1=c_1c_2\frac{r^2}{2}-c_1 \geqslant d$$
for every $j\in \mathbb{N}$ and $R\geqslant R_0$. \hfill$\Box$\smallskip
In view of the above lemmas we can now prove the existence of a suitable
sequence of approximating solutions to our
original problem (\ref{Plambda}).
\begin{proposition}\label{approximated-solutions}
Assume {\rm (\ref{H1'}), (\ref{A1}), (\ref{A2}), (\ref{G1})--(\ref{G3})},
and let $d\in \mathbb{R}$. Then there exist $R_0 >0$ and $C >0$ such that
for every $R\geqslant R_0$ the problem
\begin{equation}
-\Delta u = \lambda h(x) u + a(x) g(u), \quad u\in
H_0^1(B_R(0))\label{PlambdaR}
\end{equation}
has a solution $u_R$ satisfying $I(u_R) \geqslant d$ and $\|u_R\|_{\infty}
\leqslant C$.
\end{proposition}
\paragraph{Proof.}
We recall from section 3 that $I$ is the energy functional associated with
(\ref{PlambdaR}) (cf.
(\ref{energy-functional})). Let $R_0$ and $\ell$ be given by Lemma
\ref{estimate-from-below} and fix any $R\geqslant
R_0$. For every $j\in \mathbb{N}$ we consider the modified problem $(P)^{j}_{R}$.
In view of Lemma \ref{results-from-DR} (b),
for any $j\in \mathbb{N}$ we can choose $\rho_{\ell}^{j} > r_\ell$
($r_\ell$ as given in (\ref{definition-of-b^k})) in such a way that
$$
a_{\ell,R}^{j}:=\sup\{ I^j_R(u)\, : \, u\in V_{\ell,R} \, : \,
\|u\|=\rho_{\ell}^{j}\} \leqslant d-1.$$
Denote
\begin{gather*}
D_{\ell,R}=\{ u \in V_{\ell,R}: \, \|u\|\leqslant\rho_{\ell}^{j}\},\\
\Gamma_{\ell,R}=\{ \gamma\in C(D_{\ell,R};H_0^1(B_R(0))\, : \, \gamma
\mbox{ is odd and } \gamma|_{\partial D_{\ell,R}}=\mbox { identity}\},\\
c_{\ell,R}^{j}=\inf_{\gamma \in \Gamma_{\ell,R}}\sup_{u\in D_{\ell,R}}
I^j_R(\gamma(u)).
\end{gather*}
Since $a_{\ell,R}^{j} < b_{\ell,R}^{j}$ and since $I^j_R$ satisfies the
Palais-Smale condition (see Lemma
\ref{results-from-DR} (a)), it is known that $c_{\ell,R}^{j}$ is a critical
value for $I^j_R$ such that $c_{\ell,R}^{j}
\geqslant b_{\ell,R}^{j}$ (see e.g. \cite[Th. 5.2]{S}, \cite[Th. 3.6]{W}).
In other words, there exists
$u_{R}^{j}$ such that
$$
\nabla I^j_R (u_{R}^{j})=0 \quad \mbox{ and } \quad
I^j_R(u_{R}^{j})=c_{\ell,R}^{j}\geqslant d.$$
On the other hand, since $V_{\ell,R}$ has dimension $\ell$, we can choose
$u_R^j$ in such a way that its Morse index
is not greater than $\ell$; this follows readily from the arguments in e.g.
\cite{LS,RS}. From Lemma
\ref{results-from-DR} (c) we conclude that
$\|u_R^j\|_{\infty}$ is bounded independently of $j$. As a consequence, for
$j$ large we have that $u_R^j$ is a
solution of problem
(\ref{PlambdaR}).
At this point, for every $R\geqslant R_0$ we have constructed a solution
$u_R$ of (\ref{PlambdaR}) such that
$I(u_R)\geqslant d$. Moreover, the Morse indices $m(u_R)$ are bounded above
by some fixed number $\ell$. To finish the
proof of Proposition \ref{approximated-solutions} it remains to show that
$\|u_R\|_{\infty}$ is bounded independently of
$R$. Since $a^{+}$ has compact support, this follows as in step 2 of the
proof of Proposition
\ref{bounded-domain}.\hfill$\Box$
\paragraph{Proof of Theorem 2 completed.}
Step 1. Fix any $d \in \mathbb{R}$ and take any
sequence $R_n\to \infty$. Let $(u_n)$ be the
corresponding solutions of $(P)_{\lambda,R_n}$ given by Proposition
\ref{approximated-solutions}. As in the proof of
Theorem \ref{thm1}, up to a subsequence, $(u_n)$ has a weak limit $u$ in ${\cal
D}^{1,2}(\mathbb{R}^N)$ such that $u_n\to u$ a.\ e.\
and the sequence $(\int |a|\;g(u_n)u_n)$ is bounded. Clearly, $u$ is a
solution of (\ref{Plambda}) and $u\in
L^{\infty}(\mathbb{R}^N)$.
\noindent Step 2. By multiplying the equation in
(\ref{Plambda}) by $u\Psi_R$ where $\Psi_R$ is as in the proof of
Lemma \ref{eigenvalues} and by letting $R\to \infty$
we readily see that
$$
\int a^-g(u)u